The Loschmidt Spectral Form Factor

Abstract

The Spectral Form Factor (SFF) measures the fluctuations in the density of states of a Hamiltonian. We consider a generalization of the SFF called the Loschmidt Spectral Form Factor, $\text{tr}[e^{i{H}_{1}T}]\text{tr}[e^{-i{H}_{2}T}]$, for $H_1-H_2$ small. If the ensemble average of the SFF is the variance of the density fluctuations for a single Hamiltonian drawn from the ensemble, the averaged Loschmidt SFF is the covariance for two Hamiltonians drawn from a correlated ensemble. This object is a time-domain version of the parametric correlations studied in the quantum chaos and random matrix literatures. We show analytically that the averaged Loschmidt SFF is proportional to $e^{i\lambda T}T$ for a complex rate $\lambda$ with a positive imaginary part, showing in a quantitative way that the long-time details of the spectrum are exponentially more sensitive to perturbations than the short-time properties. We calculate $\lambda$ in a number of cases, including random matrix theory, theories with a single localized defect, and hydrodynamic theories.

16 pages, comments welcome

References in corpus (5)

• Thermalization and its mechanism for generic isolated quantum systems
• Dynamics of Loschmidt echoes and fidelity decay
• Loschmidt Echo
• Spectral Form Factor of a Quantum Spin Glass
• Fidelity and level correlations in the transition from regularity to chaos

Cited by in corpus (5)

• Scale-invariant critical dynamics at eigenstate transitions
• Many-body quantum chaos in mixtures of multiple species
• Imaginary-Temperature Zeros for Quantum Phase Transitions
• Random Circuits in the Black Hole Interior
• Leading and beyond leading-order spectral form factor in chaotic quantum many-body systems across all Dyson symmetry classes